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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | ⊢ 𝐾 = (mTC‘𝑇) |
mexval.e | ⊢ 𝐸 = (mEx‘𝑇) |
mexval.r | ⊢ 𝑅 = (mREx‘𝑇) |
Ref | Expression |
---|---|
mexval | ⊢ 𝐸 = (𝐾 × 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 ⊢ 𝐸 = (mEx‘𝑇) | |
2 | fveq2 6907 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇)) | |
3 | mexval.k | . . . . . 6 ⊢ 𝐾 = (mTC‘𝑇) | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾) |
5 | fveq2 6907 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇)) | |
6 | mexval.r | . . . . . 6 ⊢ 𝑅 = (mREx‘𝑇) | |
7 | 5, 6 | eqtr4di 2793 | . . . . 5 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅) |
8 | 4, 7 | xpeq12d 5720 | . . . 4 ⊢ (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅)) |
9 | df-mex 35472 | . . . 4 ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | |
10 | fvex 6920 | . . . . 5 ⊢ (mTC‘𝑡) ∈ V | |
11 | fvex 6920 | . . . . 5 ⊢ (mREx‘𝑡) ∈ V | |
12 | 10, 11 | xpex 7772 | . . . 4 ⊢ ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V |
13 | 8, 9, 12 | fvmpt3i 7021 | . . 3 ⊢ (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
14 | xp0 6180 | . . . . 5 ⊢ (𝐾 × ∅) = ∅ | |
15 | 14 | eqcomi 2744 | . . . 4 ⊢ ∅ = (𝐾 × ∅) |
16 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅) | |
17 | fvprc 6899 | . . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅) | |
18 | 6, 17 | eqtrid 2787 | . . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅) |
19 | 18 | xpeq2d 5719 | . . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅)) |
20 | 15, 16, 19 | 3eqtr4a 2801 | . . 3 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅)) |
21 | 13, 20 | pm2.61i 182 | . 2 ⊢ (mEx‘𝑇) = (𝐾 × 𝑅) |
22 | 1, 21 | eqtri 2763 | 1 ⊢ 𝐸 = (𝐾 × 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 × cxp 5687 ‘cfv 6563 mTCcmtc 35449 mRExcmrex 35451 mExcmex 35452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-mex 35472 |
This theorem is referenced by: mexval2 35488 msubff 35515 msubco 35516 msubff1 35541 mvhf 35543 msubvrs 35545 |
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